Download e-book for kindle: Almost Sure Convergence by William F. Stout

By William F. Stout

ISBN-10: 0126727503

ISBN-13: 9780126727500

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Extra resources for Almost Sure Convergence

Example text

1) There are three cases to consider. (i) Assume r 2 = N + 1 (no upcrossings). In this case, dN = 0 and /Ly = 0, establishing Eq. 1). (ii) Assume hN > 0 and n odd. ) In this case dN < (a — b)hN, using the fact that the number of upcrossings hN equals the number of downcrossings after the first upcrossing and that the length of each down­ crossing is less than or equal to a — b. Thus Eq. 1) holds. (iii) Assume hN > 0 and n even (last crossing an upcrossing). In this case dN < (a — b){hN — 1) + uN — utn, using the fact that the number of downcrossings after the first upcrossing equals the number of upcrossings minus one, that is, hN — 1.

Let Un = Ya=i Yt for each n > 1 define the martingale {Un, n > 1}. Let t be the first integer n > 1 such that Un > 1. Clearly EUt = 1 yet EUX = 0. Interestingly EUil) = 0 for all n > 1. 2. s. for some M < oo. Prove that E t Yt = 0. ) | Besides the optional stopping discussed above, optional sampling of a martingale (or submartingale) is possible: Let 1 < tx < t2 < • • • < tn < • • • be a sequence of stopping rules and {Un, n > 1} a martingale. Does it follow that Utl, Ut2, . . , Utn, . . is also a martingale?

The statistical problem is to decide whether {pn, n > 1} or {qn, n > 1} is the correct set of densities. 5. for each n > 1. The Tn are known as likelihood ratios. If the pn are correct, it seems plausible that the observed Yh will, when substituted into Tn, make Tn approach zero as n gets large. Likewise, if the qn are correct, it seems plausible that the observed Yh will make Tn approach infinity as n gets large. In this way, Tn, for large n, can be observed to decide whether the pn or the qn seem to be the correct densities.

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Almost Sure Convergence by William F. Stout

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